Design of High Vacuum Systems The0 ret ical and Practical Considerations C. E. Normand CARBIDE AND CARBON CHEMICALS CORPORATION, OAK RIDGE, TENN.
T h e basic formulas for the flow of gases at low pressures through pipes and apertures are reduced to forms readily applicable to the problem of vacuum system design. Methods are indicated for evaluating the conductance of channels of the more complex shapes commonly encountered in industrial systems. Factors limiting the applicability of the various methods are indicated, and the degre’e of accuracy attainable is compared with the accuracy required in design calculations.
Two basic relations follow directly from the definition of conductance: The Conductance of Series Elements. If Ct, CZ, etc., are the conductances of the various elements of a continuous flow channel, and if the temperature is constant and the pressure is stable along this channel, it follows that the quantity rate of flow is the same a t all points, and C, the net conductance of the channel, is such that 1 1 1
I”
Conductance of Parallel Elements. Under the same conditions, and for similar reasons, the joint conductance of components in parallel is
E
TE compares the problems of the modern vacuum engineer with Oh those of the vacuum technician of a decade and a half ago it is apparent that in spite of the introduction of new equip-
(3)
These relations are of basic importance in that they permit the conductance of complex systems to be calculated from the conductances of the simple components into which surh systems may be resolved. A third relation, fundamental for purppses of vacuum system design, is that relating pump speed, vacuum line conductance, and the net pumping speed of the system. The speed of a vacuum pump may be defined as the volume of gas, measured at inlet pressure, entel‘ing the pump per second. I t can then be shown that a pump of speed S pumping through a line of conductance C will remove gas from an attached chamber at a rate S’ such that
(4)
S’ may be called the net pumping speed of‘ the system, and i t is obvious that in magnitude i t can approach the speed of the pump only if the line conductance is large relative to the pump speed. It is, therefore, the problem of the vacuum engineer t o evaluate the conductance of proposed or existing vacuum lines, to the end that the net pumping speed of the system may not be too much less than the speed of the associated pump. It remains then to show how the conductance of the various typical components of which vacuum lines are composed may be evaluated in terms of the dimensions of the components, the mode of gaseous flow, and the physical properties of the gas flowing. Two distinct modes of gaseous flow are commonly encountered in vacuum systems: (1) free molecular flow, which occurs at pressures such that the molecular mean free path is of the order of the channel cross dimensions, or greater; and (2) viscous flow which occurs when the molecular mean freepath is much less than these dimensions. Because transition from either of these modes of flow to the other occurs gradually with changing pressure, there is a considerable pressure range over which a third (transition) mode of flow occum. In high vacuum systems, where pressures of less than 1 micron prevail, gaseous flow is usually free molecular. I n rough vacuum systems, on the other hand, our principal interest in line conductance and net pumping speed lies within the range of viscous flow. It is, therefore, convenient, in discussing methods of
Q Pi
+?- + . . . .
c = CI + Ca + ....
ment, materials, techniques, and processes vacuum problems remain basically unchanged. The recent increase by several orders of magnitude in the size of systems evacuated and the speed of the pumps used in their evacuation has, however, brought about changes in the importance attached to the various problems. Where small, low speed systems are involved, for example, the problems of attaining low rates of leakage, outgassing, and gas evolution are of first importance, while the problem of providing adequate vacuum line conductance can usually be solved by observance of the simple rule that all vacuum lines should be as short as possible and of the largest practical diameter. With large, high speed systems, on the other hand, much higher and often easily attained rates of leakage, outgassing, and gas evolution can be tolerated, but no simple rule is adequate for .ensuring the vary high vacuum line conductance that is necessary if a high effective rate of evacuation is to be realized. With the advent of high speed pumping systems it has therefore been found necessary to deal analytically with the gaseous flow characteristics of vacuum lines to an extent seldom required in earlier systems. A41thoughthe flow characteristics of a vacuum line may be expressed in various ways, the calculations. commonly required can most readily be made in,te‘rms of the “conductance” of the line. Conductance may be defined by the equation ,
Cx-
=
- Pa
where C is the conductance of a particular line or component, Q is the quantity of gas, in pV units flowing past any point per second, and p , - p is the pressure drop across the component. Conductance thus has the diniensions of volume per second, and it may be thought of as the volume of gas, measured at inlet pressure, that would enter the line per second if the discharge pressure were negligibly small. The conductance of a given element is, therefore, a function of the geometric dimensions of the element, and of certain physical properties of the gas flowing through it; it may or may not be a function of the pressure at which the gas flows. As a matter of fact, of the two modes of gaseous flow commonly encountered in vacuum practice, eonductance is independent of pressure in one case and directly proportional to it in the other.
783
784
calculating conductances, to consider separately the problems of high vacuum and rough vacuum conductance. Furthermore, in order that working formulas for calculating the conductance of various elements may be reduced to the most readily applicable forms, we shall consider particularly the conductance of these elements for air at 20" C., and express all quantities left standing in the formulas in the units commonly used in measuring these quantities. And finally, in order that certain approximations that are made may be justified, it should be borne in mind that conductance calculations are but the first step toward evaluating the net pumping speed of a system. If the conductance is large relative to the speed of the pump, and it certainly should be if possible, any error in the evaluation of the conductance appears as a greatly reduced error in the net pumping speed-for example, if the line conductance is twice the pump speed, an error of 10% in the conductance results in less than 3% error in the net pumping speed. Consider first the conductance to free molecular flow of various components commonly found in high vacuum lines.
Free Molecular Flow Long Pipes of Circular Cross Section. The conductance of a simple vacuum line of this type is precisely given by the formula derived and experimentally verified by Knudsen. I n its general form Knudsen's ( 5 ) equation of free molecular flow in a long pipe may be written
and, therefore
Vol. 40, No. 5
INDUSTRIAL AND ENGINEERING CHEMISTRY
(9)
and the conductance of such an aperture is
Considering again the case of air at 20" C., and expressing the area in square inches and the conductance in liters per second, we get as a working formula
C
+L
where R is the radius and L the length of the pipe, ilf is the molecular weight and T the temperature of the gas, R' is the gas constant per mole, C is the. conductance, aad all quantities are in c.g.s. absolute units. For the particular case of air a t 20" C., and with the substitution of more readily applied units, one obtains as a practical working formula
in which R is the radius and D the diameter of the pipe in inches, L is the length of the pipe in feet, and C is the conductance in
liters per second.
I n arriving a t this simple working formula certain definite limitations have been imposed.
1. The formula applies only to free molecular flow-i.e., to conditions such that the molecular mean free path is large relative to the pipe diameter. It can be shown that this will be true, and the formula -will be correct t o within 10% if
FD 5 7
(8)
where jj equals average pressure in microns, and D equals pipe diameter in inches. 2. The formula applies only to the flow of air a t 20" C. For any other gas of molecular weight, IN',and a t any other tempEa-
@
ture, T',the conductance is multiplied by a factor &. 3. The formula applies'only if the pipe is long. Any difficulty experienced by a molecule in finding the entrance t o the pipe must be negligibly small compared with the difficulty of traversing its length. This can be shown t o be true within 11%if
L (feet) 5 D (inches) 4. The pipe must be of circular cross section. For pipes and ducts of noncircular cross section the conductance is less than for pipes of circular section and equal area.
=
758
(1 1)
The Torking formula applies only to the free molecular flow oi air a t 20" C. The pressure limit for free rholecular flow and the correction required when other gases and other temperatures are involved are the same as in the cabe of long pipes. Another limitation of the aperture formula arises, however, from the assumption that the aperture area, A, is small relative to the cross-sectional area, A, of the space above it. If this is not the case a mass motion of gas toward the aperture results, arid the conductance is increased by a factor which must, obviously, assume a value of unity when >> A and approach infinity as 2 approaches A. As a suitable corrective factor Dryer (9)has given
IC=-
c = 43 fi n'T R3 AT
Small Apertures. From the kinetic theory of gases it can also be shown that the rate of free molecul~trescape of gas from a large chamber through a small aperture of area A due to a pressure difference p , - p , is
A A-A
and Loevenger (4, in an unpublished paper, has given a theoretical justification of this factor. Since h: exceeds unity by 10% or more if 2 is less than 11A, significant errors may arise from neglect of this correction in the case of large apertures. In deriving the aperture formulas no assumption is made as to the shape of the aperture, and the formula would appear to be applicable to apertures of all shapes. A question arises, however, in the case of slitlike apertures because, to molecules approaching the slit along its length, the aperture may appear large relative to both the molecular mean free path and the dimensions of the space above it, while to molecules approaching along paths perpendicular t o the length of the slit it appears small. Short Pipe of Circular Cross Section. For pipes of short length, L (feet) < D (inches), end effects are no longer negligible as assumed in deriving the long pipe formula, and the actual conductance is less than indicated by this formula. To account for this reduction Dushman (9)assumed that the conductance of a short pipe is equivalent to two conductances in series: (1) that of the pipe inlet considered a3 an aperture, and ( 2 ) that of the pipe due to its length. Combining these conductances as expressed in the working formulas previously given, one obtains as the conductance of a short pipe for air a t 20 'C. CI
75A
1
+ 9 LID
- 1
60D2
+ 9 L/D
where A equals area of pipe inlet in square inches, L equals length of pipe in feet, D equals diameter of pipe in inches, and C equals conductance in liters per second. For L = 0 this equation reduces to the aperture formula, as it should; and for L (feet) > D (inches) it differs from the long pipe formula by less than.lO%. For gases other than air and for temperatures other than 20" C. the same factor applies as in the previous cases. If the pipe entrance area is not small relative t o the area of the space above it, the large aperture correction should be applied in calculating the pipe inlet conductance. If this is done, the short pipe formula takes the form
I N D U S T R I A L A N D E N G I N E E R I N GC H E M I S T R Y
May 1948
C=A --A A
75A
+ 9L/D
(14)
To illustrate the contributions, both of the end correction as embodied in the short pipe formula and of the large aperture correction just introduced, consider the conductance of a 1-foot length of 10-inch pipe through which air is being pumped frpm the end of a cylindrical vessel 14inches in diameter. If this is treated as a long pipe, we find for its conductance
Cl = 6.5
cz
=
= -
+
'OD2 = 3160 L/sec. 1 9 L/D
60D2
A - A - +9L/D A
TRZ = ab, and R
If a / b
= 4280 Llsec.
=
4:
if
where A is the cross-section area, 0 the perimeter of the duct, and R the radius of the equivalent pipe. Both methods lead to results that are known to be only approximate, and the approximation becomes very poor in both cases for ducts of long, narrow cross section. For ducts of square or near square section it is generally assumed that the second method of calculation yields the more accurate value of conductance. Actually, the reverse is true. A comparison of the errors in the case of ducts having sections of various length to width ratios illustrates this.
% Error in Calculated Conductance R3
2 4 10
=
< 3;
treat as a round pipe of diameter D = 2
> 3;
2Az n o
- 11
15 --20
These percentages of error are based on the calculations by Clausing ( 1 ) of the conductance of rectangular ducts. Such calculations have also been made by Smoluchowski (7). Since these calculations are extremely tedious, however, they are usually
d@.
treat as a round pipe having Ds = 5.1
multiply the result by the appropriate value of K.
The second method is suggested by the fact that the RS term occurring in Iinudsen's equation arises from the division of the square of the cross-section area of the pipe by its circumference. areal Pipe and duct are then assumed t o be equivalent if perimeter is
1 2 3
The steps in calculating the conductance of a rectangular duct may be summarized:
If a / b
Neglecting the end correction leads to a positive error of some 53% and neglect of the large aperture correction results in a negative error of about 26%. Ducts of Rectangular Cross Section. For any duct of noncircular cross section therq is an equivalent round pipe having the same conductance per unit length. If the radius of this equivalent pipe can be determined from the geometry of the duct section, it is then possible t o determine the duct conductance by use of the round pipe formulas already developed. Two methods are commonly used in arriving a t values of the equivalent radius of ducts of rectangular cross section. The first method assumes that a pipe and duct will be equivalent if their cross section areas are equal-that is, if R is the radius of the pipe, and a and b are the length and width of the duct section, the two will be equivalent if
the same for both-Le.,
1.108 1.126 1.151 1.198 1.297 1.400 1,444
= 6500 Llsec. L
and with the large aperture correction applied c 3
avoided by the use of tables listing corrective factors to be applied to the results of equivalent radius calculations. Values of this factor, K , as calculated by Clausing are listed below. The &nductance as derived from the appropriate pipe formula, with 0 3 = 5.1 A2/O, is to be multiplied by the value of K corresponding to the given duct section. a/b R
0 3
If treated as a short pipe, without the large aperture correction,
785
x
A2 - and 0,
Ducts of Nonuniform Cross Section. Ducts used in joining large components are sometimes flared or tapered along their length. If the flare is uniform and symmetrical, the conductance of such ducts can be accurately calculated. Unless the flare is very great, however (dimensions across the large end equal twice the corresponding dimensions across the small end), the conductance obtained by such calculations is not significantly different from that of a uniform'duct having a cross section equal to the mean cross section of the flared duct. Flared ducts may, therefore, be treated as uniform ducts of average cross section. Channels of Indefinite Length or Irregular Cross Section. It sometimes happens that a large pipe or duct is cut obliquely a t one or both ends, with the result that no definite value can be assigned to its length; or pumping may occur through a channel whose cross section varies in no simple geometric way. In such cases it is obviously impossible to compute exact values of the conductance. It is often possible, however, by making most favorable and least favorable assumptions, t o arrive a t upper and lower limits for the conductance with s-me assurance that the true value lies somewhere between. An estimate can then he made, and in making such estimates it is well to remember that underestimates are to be preferred t o overestimates. Louvres, Baffles, Cold Traps, and Valves. Only in cases bf the simplest possible geometries can calculations of the conductances of elements such as these be made with any assured degree of accuracy. Upper and lower conductance limits can be calculated with some certainty, but in the end it is an estimated value of conductance that is assigned t o the element. Elbows and Bends. One might reasonably expect t o find very definite knowledge as to the effect of elbows and bends on the over-all conductance of high vacuum lines. ActualIy, this is far from the case. Suggested treatments range all the way from that of ignoring the bend and considering only its axial length to the assurance by one writer that a right-angle bend contributes to the length of the line its axial length plus thirty times the pipe diameter. Of these two extremes, experience indicates that the first is more nearly correct, and the majority of workers appear to agree that the effective length of a right-angle bend is its axial length plus from one to two diameters. Just what additional length is allowed for the bend, or whether any allowance a t all is made, will obviously make little difference in the total conductance of a vacuum line unless the number of bends is very great or the length of straight runs is very small.
Conductance of Rough Vacuum Lines The pressure in rough vacuum lines may range from atmospheric pressure down to a few microns, and in various parts of this pressure range all modes of gaseous flow may occur. I n most
.
INDUSTRIAL A N D E N G I N E E R I N G C H E M I S T R Y rough vacuum systhns, however, ,it is found that adequate line conductance in the viscous flow range ensures adequate conductance under all conditions. It is, therefore, the conductance of lines to viscous flow t,hat is likely to dictate the size of pipe to he used in the construction of a given rough vacuum line. Conductance of a Long Pipe to Viscous Flows. Poisseuille's ( 6 ) law of viscous flow in a long tube leads to the following simple equation for the conductance of the tube
Suppose that the speed of the pump at 45-micron intake pressure i s 70 L per second according to its volumetric efficiency curve. Then C = 9s = 630 L/sec.
is the required conductance of the 20-foot line a t 47.5-micron average pressure. Assuming that flow will be viscous under these conditions, we find from Poisseuille's law
(17) nhcre C is the conductance, R the radius and L the length of the tube or pipe, and 3 the mean piemure and 7 the viscosity of thv gas flowing. All quantities in thr above equation are expressed in c.g.b, units. Rewritten in terms of more convenient units, and with air a t 20' C. assumed as the gab, one arrives a t the working equation 1)4
-
C = 0.25 - p L
in which C equals conductance in L per second, D equals pipe diameter in inches, L equals pipe length in feet, and js equals mean pressure in microns. Unlike the corresponding case for i'ree molecular flow, in which t'he conductance for a given gas is a function of the pipe dimensions only, the conductance to viscous flow is seen to vary directly with the average pressure of the gas. 'Any value of viscous conductance applies, therefore, to some particular mean pressure. I n a given pipe. the gaseous flow is viscous, and Poisseuille's law applies, over a limited pressure range. For rough vacuum systems as actually designed, no serious error arises from assuming flow to be viscous a t all pressures below atmospheric and above that pressure at, which
pD
= 220 micron-inches
(19)
Actually, flow may become turbulent a t very high pressures, hut a t very high pressures the line conductance is so large relative t o the speed of the associated pump as to be wholly negligible. At t,he stated lower limit' it can be shown that the error from assuming viscous flow is not more t'han 10%. ,In t,he design of a rough vacuum line the first problem is that of determining an average pressure such that adequate line conductance at' this pressure will assure adequate conductance at all other pressures that may occur. Having made this choice, one may then calculate the line dimensions required to give any desired conductance at this pressure. As an example, a pumping system is to consist of a pump of 300 cubic feet per minut>edisplacement connect'ed by a 20-foot length of pipe to t,he chamber to he evacuated. This pumping system will be used to reduce the pressure from atmospheric to 50 microns, a t which point a diffusion pumping system will take over. What size of pipe should be used if the speed of pumping from the chamber is to he not less than 90% of the speed of the pump?
.
The net pumping speed will be 90% of the speed of the pump if the line conductance is equal to nine times the speed of the pump. Both the speed of the pump and the line conductance increase with increasing pressure, but conductance increases at much the greater rate. If, therefore, the line conductance is nine times the pump speed at the lowest working pressure it will be more than nine times as great a t all higher pressures. We must then determine the diameter of a 20-foot line, so that its conductance, when the inlet pressure p , = 50 microns, will equal nine times the speed of the pump under these conditions. The speed of the pump is a function of p2, the pump intake, and pipe discharge pressure, and may be determined from the volumetric efficiency curve for the pump once p2 has been determined. If C = 9s and a state of steady flow exists, then 8' = 0.9 S a n d S ' p , = Spz. Therefore, pp = 0.9 p , and since p l = 50 microns, p2 = 45 microns and p' = 47.5 microns.
Vol. 40, No. S
C
=
Di
0.25 - p
I,
I., 4 630 = '0.25 - 0.47.5 20 and
D = 5.7 inches Our assumption of viscous flow may now be checked. As PI> = 271, and flow is viscous to within 10% or bett,er for PD > 220, the assumption is valid. We would, of course, use a 6-inch pipe as the nearest stmandad size, and in so doing could be sure of 90% efficiency or better.
If we turn now t o the problem of a fore vacuum system suitable, for backing a given diffusion pump, the solution is the same exccpt for our choice of the pressure a t which the line conductance should be adjusted t o its desired value. Pressure in such a backing system varies from the base pressure of the pump, say 10 microns, up t o a value a little greater than the limiting pressure into whi(1h the diffusion pump will discharge. If the line is designed to provide t.he required conductance when its inlet pressure is equal to the limiting diffusion pump pressure, say 120 microns, the favt that the conductance yill be much less when the forepressure is only 20 microns need not concern us, as t,he fact that the forepressure is low iiitiicatm that the net forepumping speed in more than adequate. Conductance of a Short Pipe to Viscous Flow. Accurat,e corrections for the end effect in the case of viscous flow through a short pipe are difficult if not impossible to make, because the conductance of the entrance apert,ure tlo viscous flow depends not only on the entrance area but also on the shape of the aperturc and the density and velocity of t'he gas flowing t,hrough it. Limited experiments indicate, however, that end effects at pressure up t o 200 mii.rons may be approxiniately allowed for by in(-rcasing the effec.ti\repipe length by about,one diameter. Channels of Noncircular Sections, Viscous Flow. Few data are available relative t o viscous flow through channels of noncircular section, nor are such channels likely to be used in constructing rough vacuum systems. Ducts of square, or near square c r o s ~section (width to length ratios down to 0.6), may be treated a* round pipes of the same cross-section area with an accurary better t'han 90%. For lower ratios of width to length, howevei,, the conductance drops off rapidly and for a duct in which t,he u-idth of cross section is one tent'h its length thc conductance is only one quarter that of a round pipe of equal area. The work involved in calculating the approximate conductance of valves, traps, and other minor components of rough vacuum lines is seldom just,ified. A rough estimate of these conduct,ances is u s ~ a l l yadequate, as the net line conductance is many times the spwd of the associated pump and large conductance errors pi,otiuce small errors in the value of net pumping speed.
Transition Flow As has been pointed out, it is permissible to consider gascous flow as free molecular if j5D < 7 micron-inches and viwous if
ijD
> 220 micron-inches.
There remains, therefore, a tranGition
region 7
< ?;D < 220 micron-inches
within LThich flow is neither vircouS nor frw moleeulatr, arid it
May 1948
INDUSTRIAL AND ENGINEERING CHEMISTRY
sometimes happens that it is within this pressure range that our interest lies. The conductance of long pipes to the flow of gas within this pressure range is given by the semiempirical formula developed by Knudsen. Expressing the conductance for air a t 20 C., this formula may be put in the form
D4 C = 0.25 - p
L
. *
+ 6.5 -
(20)
where C equals conductance in L per second, D equals pipe diameter in inches, L equals pipe length in feet, and equals mean pressure in microns. I t is easy to show that this equation reduces to Knudsen’s equation for molecular flow when ?, is very small, and t o Poisis very large. It is seuille’s equation for viscous flow when from this equation, as a matter of fact, that the pressure limits to which these laws may be extended have been calculated. Use of the general equation in determining the conductance of a given line is tedious, but not difficult; its use in determining the diameter corresponding to a given conductance, however, is difficult. This difficulty may be resolved, for practical purposes, by recalling that the general equation need be used only over the pressure range for which ijD is greater than 7 and less than 220. Within this range the term in parentheses can never be greater than 0.84 or less than 0.81. Assigning to this term a value 0.83 is then a permissible approximation leading to the simple working formula
It is significant that the conductance throughout this transition range is greater than for either viscous or free molecular flow.
787
Thus, use of the simpler formulas up to, or even beyond, the .limits that have been set for them introduces errors that are always on the low side. One final problem of vacuum system design should be mentioned. At some point in the design of any vacuum pumping system a decision must be made as to what reduction in pumping speed due to limited line conductance can be considered reasonable or permissible. I n the case of rough vacuum lines, where pump speeds are not greater than a few hundred liters per second, where flow is viscous, and where pipes a few inches in diameter are adequate, a speed loss of 10% or less may reasonably be expected. I n high vacuum systems, however, much greater speed reduction must often be accepted. Here pump speeds of thousands of liters per second are involved, and line conductance a t free molecular flow is much lower than a t viscous flow. Unless care is taken in the design of the high vacuum system, a speed reduction of 50% or more may easily occur, and even with care it is often impractical to reduce this loss to less than 25%. If so much as a valve or baffle is placed between the pump and the evacuated chamber, it is likely that the speed reduction will exceed 10%.
Literature Cited (1) Clawing, P.,Phusica, 9,65 (1929). (2) Dryer, W.P.,C h m . Eng., 54, 127 (1947). (3) Dushman, S.,“Production and Measurement of High Vacuum,” Schenectsdy, N. Y., General Electric Co., 1922. (4) Loevenger, R., “Fundamental Considerations in Vacuum Practice,” Radiation Laboratory, Univ. of Calif., 1946 (unpublished). ( 5 ) Knudsen, M., Ann. Physik, 28,75 (1909); 35, 389 (1911). (6) Poisseuille, J. M., Compt. rend., 11, 1041 (1840); 12, 112 (1841); 15, 1161 (1842). (7) Smoluchowski, Ann. Physik, 33, 1559 (1910). RECEIVED November 3,.184,.
New Techniques in the Measurement of Pressures below 10 Mrn. Glenn L. Mellen
NATIONAL RESEARCH CORPORATION, CAMBRIDGE, MASS.
T h e general problem of vacuum measurement and control for pressures below 10 mm. is described. Some indication is given of the limitations of conventional type gages and a discussion presented of representative techniques for utilizing intelligence from gages for control purposes. The radium source type of gage in various forms is mentioned with indications of its uses and limitations.
T
HE common units of pressure measurement in a vacuum are the millimeter and micron of mercury (1 mm. = 1000 microns). Pressure is defined as the height of a column of mercury that can be supported by the unknown pressure if a zero pressure exists above the column. It may be measured either by. some gage that indicates the time rate of molecular momentum transfer directly (McLeod gage, Knudsen manometer) or one that measures a pressure-dependent property of the gas (5) (Figure 1). Of these properties, heat conductivity and molecular ionization are the most widely used. Pirani and thermocouple gages depend upon the conductivity phenomenon,
while hot filament and radioactive source gages depend upon the ionization phenomenon, Of the first group (direct indicating) the limit of sensitivity is usually dictated by two natural torces: atmospheric pressure and gravity. These gages have necessary constructional details that cause appreciqble hysteresis effects at low pressures and thus determine their lower limit of usefulness. For example, differential oil manometer can be constructed with immiscible oils of almost the same specific gravity. The calculated length of a 100-micron pressure scale may be several inches, but the adhesion of the oil to the walls of such a gage may well give a 10- to 20-micron hysteresis error. Gages of this group will not be discussed further here. The gages in the second group have different pressure sensitivities for different gases and therefore require a calibration factor for each gas, commonly referred to as air unity. Thus for a radium-source ionization gage, we have a group of curves such as those shown in Figure 2. In general, all ionization gages have linear calibration and the magnitude of their ion currents is a function of the density and the energy of the ionizing agent as
